Problem 5
Prove that (a squared plus 2)(b squared plus 2)(c squared plus 2) is at least 9(ab+bc+ca) for all positive real numbers a,b,c.
Step 3 of 5: Reduce the discriminant to a second quadratic
Detailed analysis
A direct expansion of the discriminant of F gives the displayed expression. The first factors are positive, so it is enough to prove G(b)=(2c squared+1)b squared−6cb+c squared+2 is nonnegative.